Unorientable Surfaces

Unorientable surfaces are commonly referred to in mathematical
popularizations as one-sided surfaces. The simplest
example is the Möbius strip, and you're often asked to imagine
what would happen if you tried to paint it different colors on opposite
sides:

Alternatively you are asked to imagine what it would be like to be
an ant crawling upon the surface of a Möbius strip, as
in this famous work of the Dutch artist Maurits Escher:

However all of these views of the Möbius strip are extrinsinsic:
they look upon the Möbius strip as viewed from outside from surrounding
3-dimensional space. In contrast the point of view that modern topology
takes with respect to the Möbius strip is the point of view of a mythical
2-dimensional being imbedded entirely within the Möbius strip.
Think of a 2-dimensional crab swimming in a 2-dimensional aquarium having the
shape of a Möbius strip:

Notice that as the crab completes a circuit around the Möbius strip,
its right and left sides gets interchanged. This is the crucial feature
of norientable surfaces which distinguishes them from the more familiar
orientable "two-sided" surfaces.

Once we adopt this point of view, we can use a much more convenient model
of the Möbius strip:

This "gluing diagram" asks us to take a rectangle and abstractly
glue the left hand side to the right hand side with a twist. We emphasize
the word "abstractly", ie. we are not supposed to move the
rectangle around, physically twist it in three dimensional space and glue it.
Rather we should imagine this from the point of view of our 2-dimensional crab
moving within this abstractly glued rectangular strip. As the crab moves rightwards
and encounters the right edge, it starts to emerge at the left edge, but with a twist,
because of the abstract gluing. A similar thing happens when the crab moves leftwards
and encounters the left edge: it then reemerges at the right edge, again with a
twist. (Note that the crab doesn't notice the right or left edge as it crosses.
From its point of view these edges aren't there: it's just the same to the crab as
moving around in the middle of the rectangle. Nor does the crab notice its left
and right sides being interchanged, until it returns to its starting point.)
However when the crab encounters the top or bottom edge, it bounces
back, since these edges aren't glued to anything else -- they form the boundary
of the Möbius strip. (Technically this process of abstract gluing is called
forming a quotient space.) We can easily simulate this using computer graphics:

The Möbius strip is named after the German mathematician
August Möbius (1790-1868).

We can now use the insights we have acquired here to study
another more exotic example of an unorientable surface,
the Klein bottle.